The Andrássy Rifts

The villas along the section of Andrássy Avenue between Kodály Körönd and Heroes' Square almost all have uncontrolled portals into various parts of the Umbra

There, I said it.
This is arguably the backbone of the setting - it was the first of the defining ideas.

Let me flesh this out.
Andrássy Avenue (Andrássy út) is a UNESCO World Heritage site, and for good reason - it is the lined with grand palazzos on both sides, and a wide, 1.5-mile promenade (longer than the Champs-Élysées!) with an antique underground line rumbling just below street level.
The villas and palazzos in this alternate reality have all been owned by grand Hermetic masters and their retinue, passed down through generations. No villa of a Hermetic wizard could do without a portal - I think that's understandable.
When the Hermetics were ousted from power around 1989, the inhabitants of these grand villas have fled, and the portals fell into disuse. Without maintenance, these portals have started to seep - Umbrood, spirits, talismans and other magic items, and Horizon Realm denizens have been drifting through the Gauntlet into the heart of Budapest.

This is either menacing or liberating - whatever your plot/themes require

You (the GM and/or the troupe) can treat this region as a Shallowing or a part of the city where vulgar magic is more likely to be passed off as coincidental.
However, my original intention was to

• enhance diversity of themes and types of plots -> basically, there is no need to go anywhere else out of Budapest, just have whatever you need come through a portal)
• enhance diversity of PCs and NPCs - the Andrássy Rifts attract a vast number of "tourist" mages from all over the world; any and every sort of PC/NPC mage may be justified to be milling around in Budapest, and there is a really big underground market of magical items if you know where to look
• offer opportunities for Technocrat-Traditionalist co-operation - to fight off some Nephandic threat that slipped through unheeded
• provide a source of wealth for the city (through tourism and the grey magical market) that the powers that be could cite as their political achievement and exploit to stay in power
• provide a source of 'terrorist threat' that the powers that be can cite as something they're defending the city from, once again leveraging it to stay in power

Basically, Andrássy Avenue = Bondi Beach

Just replace "laid-back surfer dudes with strange haircuts and jewellery milling about in just shorts and wetsuits" with "exotic and weird wizards of every stripe doing strange stuff".
Andrássy Avenue is a magnet for a particular kind of mage - there's Tass everywhere, and the adrenaline rush of uncontrolled Umbrood is just the sort of excitement a 'surfer mage' gets a kick out of.
There is probably a cottage industry around mage tourists - buskers, cafés, guided haunted villa tours.
This place is basically a licence to go crazy: It could be the Dreaming from the original Changeling just as much as the Zone from Stalker.